Rethinking and researching the physical meaning of the standard linear solid model in viscoelasticity

Abstract

Despite the common use of the standard linear solid model (SLSM) in viscoelasticity, the physical significance as well as the difference between the Maxwell and Kelvin forms of SLSM are still not clear. This paper demonstrates that each parameter of those two models has its specific physical meaning, and introduces the relationships allowing the transformations between those parameters. Regardless of their physical significance, those two models are equivalent in terms of their mathematical properties. Hence, no matter which model is chosen, consistent analysis results can always be obtained as long as the physical meaning of each parameter is accurately interpreted.

Keywords:

• Stress relaxation
• creep
• hysteresis
• Maxwell model
• Kelvin model
• viscoelastic properties

1. Introduction

Most materials, including those found in nature and manufactured artificially, are viscoelastic [1–8]. It means that most materials possess mechanical behaviors similar to those of viscous fluids and elastic solids [3, 9–15], and display time-dependent viscoelastic behaviors such as stress relaxation and creep when they are subjected to loading [9, 16–23]. Through measuring and analyzing the viscoelastic behaviors of a material, the viscoelastic properties of that material can be quantitatively evaluated [11, 22, 24–30]. Quantitative evaluation of the viscoelastic properties of materials is valuable for numerous applications in different fields. For example, in mechanical engineering, the viscoelastic properties of materials and structures are important parameters determining the mechanical behaviors and performances of materials, structures and the entire system, therefore an accurate evaluation of those properties plays an important role in optimal design of mechanical systems [21, 25, 27, 28, 31–39]; in civil engineering, the ability to evaluate the changes in the viscoelastic properties of asphalt mixtures and pavement structures caused by aging is beneficial to quantitatively monitor the effects of aging on their conditions [40–44]; in the field of vibration control and damping technology, the viscoelastic properties of damping materials in mechanical machines and building structures determine the damping ability of such materials to reduce vibrations for avoiding failure due to fatigue [45–47]; in polymer science and engineering, the viscoelastic properties of polymers are often used to classify and compare polymers as well as to quantify the functional characteristics of polymers, owing to the close relationship between the viscoelastic properties, composition, structure and quality of polymers [48–51]; in food science, the viscoelastic properties of food can be used to quantify and ensure the quality of food during manufacturing, processing and transportation in the food chain [14, 52–54]; in biomedical engineering, the viscoelastic properties of engineered tissues and biomaterials can be used to understand the conditions of such materials during the design process, facilitating to develop materials for restoring the functionality of damaged tissues or achieving certain functional outcomes [22, 55–63]; in clinical medicine, the viscoelastic properties of tissues could be served as diagnostic biomarkers for differentiating normal and pathological tissues as well as for quantifying the severity of injuries and diseases, since injuries and diseases may cause significant changes in the viscoelastic properties of tissues [24, 29, 30, 64–71]; similarly, an understanding of the viscoelastic properties of cells could provide insights into the mechanisms and evolution of a disease, since the viscoelastic properties of cells could be changed significantly during the disease process [72–77]. The above-mentioned examples are just some of the numerous applications where the evaluation of the viscoelastic properties can be valuable. In order to achieve an accurate evaluation of the viscoelastic properties of materials, it is important to choose an appropriate viscoelastic model for the evaluation and to completely understand the properties of that model.

In literature, various kinds of viscoelastic models have been proposed to describe and analyze viscoelastic behaviors [9, 14, 16–19, 69, 78–80], including physical-based models (that is, models constructed based on the physical mechanisms of materials) [81–86] and fractional-order models [8, 11, 14, 35, 38, 46, 52, 71, 87–91]. However, to date, the most well-known and widely used viscoelastic models are still the classical models constructed using linear springs and linear dashpots (abbreviated as spring and dashpot respectively hereafter), such as the Maxwell model, Kelvin model, standard linear solid model, Burger’s model, generalized Maxwell and Kelvin models, and so on [92]. In such models, springs and dashpots are used to model the mechanical behaviors of elastic solids and viscous fluids respectively, and their parameters represent the viscoelastic properties of materials being modeled. To evaluate the viscoelastic properties of a material, a specific kind of viscoelastic behavior (for example, stress relaxation or creep) of that material is experimentally measured, then the associated solution of a chosen viscoelastic model for that kind of viscoelastic behavior is used to curvefit the experimentally measured viscoelastic behavior to extract the parameters of the model for quantifying the viscoelastic properties of that material.

In the viscoelastic models constructed using springs and dashpots, the generalized Maxwell and Kelvin models are considered to be the most general, but they have some limitations. The generalized Maxwell model can only be used to analyze stress relaxation, and is not suitable for analyzing creep. On the other hand, the generalized Kelvin model can only be used to analyze creep, and is not suitable for analyzing stress relaxation. Both of them are not suitable for analyzing hysteresis (another typical viscoelastic behavior that is often discussed). In addition, the solution of the generalized Maxwell model for describing stress relaxation and the solution of the generalized Kelvin model for describing creep are typically derived based on the assumption that the loading is a step function, implying that the rate of loading is infinite (that is, extremely fast). Using such solutions derived with infinite loading rate to analyze experimental data may cause significant errors, since the loading rate must be finite in real stress relaxation and creep experiments, no matter how fast it is. Last but not least, although a generalized Maxwell or Kelvin model always can successfully curvefit a set of data so long as it consists of a large enough number of springs and dashpots (that is, its solution form consists of a large enough number of exponential functions, since its solution is of the form of a linear combination of exponential functions, with the number of exponential functions is equal to that of dashpots in the model), a generalized Maxwell or Kelvin model with a large number of springs and dashpots is actually not a physical model but just a purely mathematical model for analyzing data, since the physical meanings of such a large number of springs and dashpots are no longer significant.

The standard linear solid model (Figure 1), a three-element model consisting of two springs and a dashpot, is one of the most common and popular viscoelastic models used to describe and analyze viscoelastic behaviors of viscoelastic solids. It can be used to simulate viscoelastic behaviors for understanding rich phenomena and underlying mechanisms of viscoelastic behaviors, and also can be used to curvefit experimentally measured (or computationally simulated) viscoelastic behaviors for evaluating the viscoelastic properties of materials. The standard linear solid model has two forms: the Maxwell form (Figure 1a, abbreviated as SLSM_Maxwell hereafter), in which a spring is in parallel connection with a Maxwell model (a two-element model consisting of a spring and a dashpot in series), and the Kelvin form (Figure 1b, abbreviated as SLSM_Kelvin hereafter), in which a spring is in series connection with a Kelvin model (a two-element model consisting of a spring and a dashpot in parallel). SLSM_Maxwell and SLSM_Kelvin are two different models, even though they look similar since both of them are constructed by two springs and a dashpot. To our best knowledge, in literature, SLSM_Maxwell and SLSM_Kelvin are not specifically distinguished, and researchers would choose one of them to use without specific reasons. However, they are actually two different models.

Figure 1. (a) The Maxwell form of the standard linear solid model. $E_1,$ $E_2$ and $\eta$ are the parameters of this model relevant to the viscoelastic properties. (b) The Kelvin form of the standard linear solid model. $E_1^{\prime },$ $E_2^{\prime }$ and ${\eta }^{\prime }$ are the parameters of this model relevant to the viscoelastic properties.

The standard linear solid model has several advantages. First, its solutions can be derived using any forms of loadings (for example, a ramp loading with an arbitrary finite loading rate, a loading with infinite loading rate, a sinusoidal loading, an arbitrary loading, etc.), therefore the standard linear solid model can describe all possible viscoelastic behaviors. Second, the standard linear solid model is simple since it has only three elements, therefore its analysis results are much easier to be interpreted compared to those using models having a large number of elements (such as the generalized Maxwell and Kelvin models). Third, the mathematical form of the standard linear solid model has the potential to be modified, so that its ability to curvefit data and to realistically quantify viscoelastic properties, as well as its physical significance can be further improved. In literature, the standard linear solid model has been widely used to describe and analyze the viscoelastic behaviors of numerous materials and structures such as mechanical elements [93–97], polymers [98–109], hydrogels [110–113], cells [114–119], and biological tissues including blood vessels [120, 121], heart muscles [122], cartilages [123, 124], intervertebral disks [125, 126], to name a few.

Despite the common use and popularity of the standard linear solid model, the physical meanings of its parameters are still not clear. In literature, typically, the physical meanings of its parameters are not or just simply mentioned, and have not been clearly interpreted and rigorously proved. There are more questions to be answered about the standard linear solid model, and the answers of these questions are important not only fundamentally but also practically. For example, is there any difference between the two springs of a form of the standard linear solid model? Is there any difference between the two springs of SLSMMaxwell and the two springs of SLSMKelvin, and between the dashpot of SLSMMaxwell and the dashpot of SLSMKelvin? Furthermore, what is the difference between SLSM_Maxwell and SLSM_Kelvin, and which one should be chosen to use? The answers for those questions are extremely important for accurately evaluating the viscoelastic properties of materials, as well as for accurately interpreting the analysis results and optimizing the application outcomes. Hence, the purpose of this paper is to clearly demonstrate the physical meaning of each parameter of SLSM_Maxwell and SLSM_Kelvin, and to clarify the relationship and difference between those two forms of the standard linear solid model.

2. Investigating the physical meanings of the parameters of the Maxwell form of the standard linear solid model

2.1. Overview

In this section, the theoretical background for demonstrating the physical meanings of the parameters of SLSM_Maxwell will be presented. Three different perspectives will be respectively used for the demonstration, and it can be seen that those three different perspectives will lead to the same conclusion. One perspective is based on the viscoelastic behaviors in the loading process. The other two perspectives are based on the viscoelastic behaviors in the stress relaxation and creep processes . Please see Part A of the supplementary material for an introduction to the loading and stress relaxation processes in the stress relaxation test, and Part B of the supplementary material for an introduction to the loading and creep processes in the creep test. Stress relaxation and creep tests are two conventional and standard methods for evaluating the viscoelastic properties of materials. In literature, some novel techniques for this purpose have been proposed. For example, a novel analytical technique termed i-Rheo for evaluating the materials’ linear viscoelastic properties over the widest possible range of frequencies has been proposed [74, 127, 128]. The main advantage and feature of this novel technique are that it is simple and free from prescribed assumptions and models, and its evaluation is based on analyzing the Fourier transforms of experimental data involving both the time-dependent stress and strain functions excited using a simple step strain input. This technique can be widely used to analyze data obtained through various apparatuses, such as conventional material testing systems and atomic force microscopy [74, 128]. The author of the present paper believes that this kind of new concept can also be useful to demonstrate the physical meanings of classical viscoelastic models such as the standard linear solid model. In this paper, the conventional loading, stress relaxation and creep processes will be used for the demonstration.

Subsection 2.2 will demonstrate the physical meanings of the parameters of SLSM_Maxwell based on the perspective of the loading process. Subsection 2.3 will do the demonstration based on the perspectives of the stress relaxation and creep processes.

2.2. From the perspective of the loading process for demonstrating the physical meanings of the parameters of the Maxwell form of the standard linear solid model

In this subsection, the physical meanings of $E_1$ and $E_2$ of SLSM_Maxwell will be demonstrated based on the perspective of the loading process, while the physical meaning of $\eta$ of SLSM_Maxwell will be explained based on the model structure of SLSM_Maxwell.

The constitutive equation of SLSM_Maxwell is: (1) $$\sigma +{\tau }_R\dot{\sigma }=E_1\left(\epsilon +{\tau }_C\dot{\epsilon }\right)$$(1) where $\sigma$ and $\epsilon$ are the stress and strain as functions of time, while ${\tau }_R=\eta /E_2$ and ${\tau }_C=\eta \left(E_1+E_2\right)/\left(E_1{E}_2\right)$ are the relaxation and creep time constants described by SLSM_Maxwell, respectively. $E_1,$ $E_2$ and $\eta$ are the parameters of SLSM_Maxwell relevant to the viscoelastic properties. Please see Part C of the supplementary material for the derivation of Eq. (1)(1) $$\sigma +{\tau }_R\dot{\sigma }=E_1\left(\epsilon +{\tau }_C\dot{\epsilon }\right)$$(1) .

Using SLSM_Maxwell, the stress-strain relationship of a viscoelastic solid in the loading process (that is, the solution of SLSM_Maxwell for describing the loading process) can be described by the following equation, which can be derived from Eq. (1)(1) $$\sigma +{\tau }_R\dot{\sigma }=E_1\left(\epsilon +{\tau }_C\dot{\epsilon }\right)$$(1) : (2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) where ${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)$ is the stress in the loading process as a function of $\epsilon ,$ while $r$ is the strain rate used in the loading process that is the first derivative of strain with respect to time (that is, $r=d\epsilon /\mathit{\text{dt}}$). Please see Part D of the supplementary material for the derivation of Eq. (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) .

Equation (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) is a nonlinear equation with respect to strain. It means that, in the loading process, a viscoelastic solid may have different ratios of stress to strain (that is, the slope of the stress-strain curve evaluated at a strain value) at different strains.

It will be shown soon below that the physical meanings of both $E_1$ and $E_2$ are associated with certain kinds of moduli of elasticity. Hence, in the following demonstration, the equation for describing the modulus of elasticity of a viscoelastic solid obtained through the loading process will be derived first. In this paper, the secant modulus of elasticity, calculated by the slope of the line between the initial and last points of the stress-strain curve of the loading process, is used to define the “measured” modulus of elasticity of a viscoelastic solid obtained through the loading process. Hence, the equation for describing the measured modulus of elasticity is: (3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) where ${\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)=E_1{\epsilon }^{\prime }+E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right),$ described using Eq. (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) . ${\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)$ and ${\epsilon }^{\prime }$ are the stress and strain at the last point of the loading curve, respectively. In other words, ${\epsilon }^{\prime }$ is the maximal strain in the loading process, and ${\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)$ is the corresponding stress at that strain. ${\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)$ and ${\epsilon }_{t=0}$ are the stress and strain at the initial point of the loading curve respectively, and they are both equal to zero.

Please note that, as opposed to the “intrinsic” modulus of elasticity which is an inherent material property, the “measured” modulus of elasticity is a quantity measured by a measurement system (such as a material testing system) in an experiment. The measured modulus of elasticity may not be equal to the intrinsic modulus of elasticity, since the value of the measured modulus of elasticity depends on several factors such as the strain rate used in the measurement. Generally, more or less differences (or called errors) exist between the measured modulus of elasticity and the intrinsic one.

There are two terms in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) : $E_1$ and $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime }.$ Since Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) is the description of the measured modulus of elasticity obtained through the loading process, the physical meaning of each of its two terms must be associated with a certain kind of modulus of elasticity. Compositionally, a viscoelastic solid can be regarded as composed of two material parts: elastic solid part and fluid part. Hence, it can be reasonably inferred that one term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) should be associated with the modulus of elasticity of the elastic solid part, and the other term is associated with the modulus of elasticity of the fluid part.

The first term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_1,$ is a constant independent of the strain rate $r.$ Since it is known that the mechanical response of an elastic solid is independent of the strain rate, the physical meaning of $E_1$ must be the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid. On the other hand, the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime },$ is dependent on the strain rate $r.$ Since it is known that the mechanical response of a fluid is dependent on the strain rate, the physical meaning of this term must be associated with a certain quantify of the fluid part of a viscoelastic solid. In order to further confirm the physical meaning of the term $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime },$ let’s investigate how Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) changes with the strain rate $r.$ Two extreme values of the strain rate $r$ (that is, $r\to 0$ and $r\to \mathrm{\infty }$) will be investigated.

In the first case when $r\to 0,$ the measured modulus of elasticity is: (4) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1$$(4) Please see Part E of the supplementary material for the derivation of Eq. (4)(4) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1$$(4) . It can be shown that $E_1$ is the minimal measured modulus of elasticity that can be measured for a viscoelastic solid through the loading process occurred when $r\to 0$ (please see Part E of the supplementary material for the proof), as illustrated in Figure 2. Equation (4)(4) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1$$(4) suggests that, when the strain rate is extremely slow, the measured modulus of elasticity is approximately equal to the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid, $E_1.$ The first term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_1,$ is a constant independent of the strain rate, therefore this term is always $E_1$ regardless of the strain rate. The second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime },$ is zero when the strain rate is extremely slow. It means that this term has no effect on the measured modulus of elasticity when the strain rate is extremely slow. Since this term is associated with the fluid part of a viscoelastic solid, it means that the fluid part of a viscoelastic solid exhibits no stiffness and contributes nothing to the measured modulus elasticity when the strain rate is extremely slow.

Figure 2. Illustration of the relationship between the measured modulus of elasticity ($E_{\mathit{\text{measured}}}$) and the strain rate ($r$). $E_1$ and $E_1+E_2$ are the minimal and maximal measured moduli of elasticity that can be measured for a viscoelastic solid through the loading process occurred when $r\to 0$ and $r\to \mathrm{\infty },$ respectively.

In the second case when $r\to \mathrm{\infty },$ the measured modulus of elasticity is: (5) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1+E_2$$(5) Please see Part E of the supplementary material for the derivation of Eq. (5)(5) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1+E_2$$(5) . It can be shown that $E_1+E_2$ is the maximal measured modulus of elasticity that can be measured for a viscoelastic solid through the loading process occurred when $r\to \mathrm{\infty }$ (please see Part E of the supplementary material for the proof), as illustrated in Figure 2. Equation (5)(5) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1+E_2$$(5) suggests that, when the strain rate is extremely fast, the measured modulus of elasticity is equal to $E_1+E_2.$ The first term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_1,$ is a constant independent of the strain rate, therefore this term is always $E_1$ regardless of the strain rate. The second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime },$ is $E_2$ when the strain rate is extremely fast. This term is a strictly increasing function of $r$ at the open interval $0<r<\infty ,$ and it means that, the larger $r,$ the larger the value of this term, and zero is minimum while $E_2$ is maximum of this term (please see Part E of the supplementary material for the proof). It means that this term has the maximal effect on the measured modulus of elasticity when the strain rate is extremely fast. Since this term is associated with the fluid part of a viscoelastic solid, it means that the fluid part of a viscoelastic solid exhibits the maximal stiffness and contributes the most to the measured modulus elasticity when the strain rate is extremely fast.

Based on the theory of continuum mechanics [129], the amount of energy transferred to an incompressible Newtonian fluid (or called linear fluid) per unit volume per unit time during loading is: (6) $$P=2\eta r^2$$(6) where $\eta$ is the viscosity of the incompressible Newtonian fluid, and $r$ is the strain rate used in the loading process. There are two observations from examining Eq. (6)(6) $$P=2\eta r^2$$(6) . First, $P$ is zero when $r\to 0,$ and it means that there is no energy transferred to the incompressible Newtonian fluid if the strain rate is extremely slow. This observation can explain the finding that the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) associated with the fluid part of a viscoelastic solid is zero and has no effect on the measured modulus of elasticity when the strain rate is extremely slow. The second observation is that, $P\to \mathrm{\infty }$ when $r\to \mathrm{\infty }.$ It means that the amount of energy transferred to the incompressible Newtonian fluid is maximum and the energy lost is minimum if the strain rate is extremely fast. This observation can explain the finding that the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) associated with the fluid part of a viscoelastic solid has the maximal value (that is, $E_2$) and contributes the most to the measured modulus of elasticity when the strain rate is extremely fast. Since the amount of energy transferred to the incompressible Newtonian fluid is maximum and the energy lost is minimum when $r\to \mathrm{\infty },$ the properties of the incompressible Newtonian fluid can be fully measured without any loss when $r\to \mathrm{\infty }.$

Based on the analysis above, the physical meaning of $E_2$ can be summarized as below. Since the physical meaning of the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) is associated with the modulus of elasticity of the fluid part of a viscoelastic solid, its maximal value without any loss, $E_2,$ must be the intrinsic modulus of elasticity of the fluid part of a viscoelastic solid. Only when the strain rate is extremely fast, the fluid part of a viscoelastic solid can fully exhibit its intrinsic modulus of elasticity $E_2$ as the measured modulus of elasticity without any loss; therefore, the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) approaches $E_2$ when the strain rate $r$ approaches infinity. On the other hand, the smaller the strain rate, the larger the energy loss, and the lower the measured modulus of elasticity can be exhibited by the fluid part of a viscoelastic solid; therefore, the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) approaches zero when the strain rate $r$ approaches zero.

The analysis above also implies the experimental method for testing the intrinsic moduli of elasticity of the elastic solid part and fluid part of a viscoelastic solid. The intrinsic modulus of elasticity of the elastic solid part, $E_1,$ is approximately equal to the measured modulus of elasticity tested when the strain rate is extremely slow. On the other hand, the intrinsic modulus of elasticity of the fluid part, $E_2,$ is approximately equal to the difference between the measured modulus of elasticity tested when the strain rate is extremely fast (that is, $E_1+E_2$) and the measured modulus of elasticity tested when the strain rate is extremely slow (that is, $E_1$). In practice, the theoretical values of $E_1$ and $E_2$ cannot be perfectly obtained and only can be approximately measured, since the strain rate cannot be zero or infinity in a real experiment.

The model structure of SLSM_Maxwell also has significant physical meanings (Figure 3). Since SLSM_Maxwell is a model consisting of a spring and a Maxwell model in parallel, in which spring and Maxwell model are known as elastic solid model and fluid model respectively, it can be intuitively realized that the spring and Maxwell model of SLSM_Maxwell represent the elastic solid part and fluid part of a viscoelastic solid respectively (Figure 3). Since the dashpot of SLSM_Maxwell is an element within the Maxwell model (a fluid model, as mentioned above), it can be intuitively and reasonably realized that the physical meaning of the parameter of the dashpot $\eta$ is the viscosity of the fluid part of a viscoelastic solid.

Figure 3. SLSM_Maxwell is a model consisting of a linear spring and a Maxwell model in parallel, in which linear spring and Maxwell model are known as elastic solid model and fluid model respectively. Hence, the linear spring and Maxwell model of SLSM_Maxwell represent the elastic solid part and fluid part of a viscoelastic solid respectively.

In summary, based on the perspective of the loading process, the physical meanings of the parameters of SLSM_Maxwell can be demonstrated: $E_1$ is the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid, and $E_2$ is the intrinsic modulus of elasticity of the fluid part of a viscoelastic solid. The physical meaning of $\eta$ is the viscosity of the fluid part of a viscoelastic solid, explained based on the model structure of SLSM_Maxwell.

Let’s provide some further information before closing this subsection. It has been explained above that, the first term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_1,$ is a constant independent of $r,$ and the second term in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , $E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)/{\epsilon }^{\prime },$ is a strictly increasing function of $r$ at the open interval $0<r<\infty .$ Hence, the addition of those two terms, that is, Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) , the description of the measured modulus of elasticity obtained through the loading process $E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right),$ is a strictly increasing function of $r$ at the open interval $0<r<\infty .$ It means that, the larger $r,$ the larger $E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right).$ Since $E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)$ is a strictly increasing function of $r$ at the open interval $0<r<\infty$ while $\underset{r\to 0}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=E_1$ and $\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=E_1+E_2,$ it can be known that $E_1$ and $E_1+E_2$ are minimum and maximum of $E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right),$ respectively. In other words, $E_1$ and $E_1+E_2$ are the minimal and maximal measured moduli of elasticity that can be measured for a viscoelastic solid occurred when $r\to 0$ and $r\to \mathrm{\infty },$ respectively.

Some simulation results for explaining the theoretical analysis proposed above are shown below. The simulation and data analysis are performed using MATLAB (R2021a, Mathworks, Natick, Massachusetts, USA). In this simulation, nine materials with different viscoelastic properties are investigated, as shown in Table 1. The chosen values of the modulus of elasticity [130] and the relaxation time constant [131] of those materials are within the range of reported values for soft tissues. For each material, the measured moduli of elasticity at seven strain rates (10⁻⁶, 10⁻³, 10⁻², 10⁻¹, 1, 10 and 10⁴ 1/s, from extremely slow for mimicking $r\to 0$ to extremely fast for mimicking $r\to \mathrm{\infty }$) are simulated using Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) . ${\epsilon }^{\prime }$ in Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) is set as a constant of 0.05. To provide an example for illustration, Figure 4(a) shows a set of stress-strain curves in the loading process simulated using Eq. (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) for the material No.1, and Figure 4(b) shows the relationship between the measured modulus of elasticity and the strain rate simulated using Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) for that material. It can be observed that the simulation results shown in Table 1 and Figure 4 can explain the theoretical analysis proposed above, that:

Figure 4. Simulation results associated with Table 1 as an example for illustration: (a) a set of stress-strain curves in the loading process simulated using Eq. (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) for the material No.1; (b) the relationship between the measured modulus of elasticity and the strain rate simulated using Eq. (3)(3) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}$$(3) for that material.

Table 1. Simulation results regarding the measured moduli of elasticity at different strain rates obtained through the loading process, for nine materials with different viscoelastic properties.

Material number	Setting of the viscoelastic properties			Measured modulus of elasticity (Pa)
				Strain rate (1/s)
	$E_1$ (kPa)	$E_2$ (kPa)	$\eta$ (kPa $·$ s)	10^-6	10^-3	10^-2	10^-1	1	10	10^4
1	5	20	10	5000.2	5200.0	6999.9	17,642.4	24,032.5	24,900.3	24,999.9
2	5	20	40	5000.8	5800.0	12,343.3	22,695.9	24,752.1	24,975.0	25,000.0
3	5	20	100	5002.0	6999.9	17,642.4	24,032.5	24,900.3	24,990.0	25,000.0
4	10	40	20	10,000.4	10,400.0	13,999.8	35,284.8	48,065.0	49,800.7	49,999.8
5	10	40	80	10,001.6	11,600.0	24,686.6	45,391.9	49,504.1	49,950	49,999.9
6	10	40	200	10,004.0	13,999.8	35,284.8	48,065.0	49,800.7	49,980.0	50,000.0
7	30	120	60	30,001.2	31,200.0	41,999.5	105,854.5	144,195.1	149,402.0	149,999.4
8	30	120	240	30,004.8	34,800.0	74,059.9	136,175.6	148,512.4	149,850.1	149,999.8
9	30	120	600	30,012.0	41,999.5	105,854.5	144,195.1	149,402.0	149,940.0	149,999.9

1. The measured modulus of elasticity approaches $E_1$ when $r\to 0.$

2. The measured modulus of elasticity approaches $E_1+E_2$ when $r\to \mathrm{\infty }.$

3. The measured modulus of elasticity is a strictly increasing function of $r$ at the open interval $0<r<\infty .$ $E_1$ and $E_1+E_2$ are the minimal and maximal measured moduli of elasticity that can be measured for a viscoelastic solid occurred when $r\to 0$ and $r\to \mathrm{\infty },$ respectively.

2.3. From the perspective of the stress relaxation and creep processes for demonstrating the physical meanings of the parameters of the Maxwell form of the standard linear solid model

In this subsection, the physical meanings of $E_1$ and $E_2$ of SLSM_Maxwell will be demonstrated based on the perspectives of the stress relaxation and creep processes. Please note that, the physical meaning of $\eta$ of SLSM_Maxwell has been explained based on the model structure of SLSM_Maxwell in the last subsection, therefore will not be demonstrated in this subsection.

It will be shown soon below that the physical meanings of both $E_1$ and $E_2$ are associated with certain kinds of moduli of elasticity. Hence, in the following demonstration, the equation for describing the relaxation modulus of a viscoelastic solid in the stress relaxation process will be derived first. The relaxation modulus is a strictly decreasing function of time that describes how the modulus of elasticity of a viscoelastic solid decreases over time in the stress relaxation process.

Using SLSM_Maxwell, the stress-time relationship of a viscoelastic solid in the stress relaxation process (that is, the solution of SLSM_Maxwell for describing the stress relaxation process) can be described by the equation: (7) $${\sigma }_{\mathit{\text{relaxation}}}\left(t\right)=\left(E_1+E_2{e}^{\frac{-t}{{\tau }_R}}\right){\epsilon }_0$$(7) where ${\sigma }_{\mathit{\text{relaxation}}}\left(t\right)$ is the stress in the stress relaxation process as a function of time $t,$ and ${\epsilon }_0$ is the constant strain in the stress relaxation process. Please see Part G of the supplementary material for the derivation of Eq. (7)(7) $${\sigma }_{\mathit{\text{relaxation}}}\left(t\right)=\left(E_1+E_2{e}^{\frac{-t}{{\tau }_R}}\right){\epsilon }_0$$(7) . In order to derive Eq. (7)(7) $${\sigma }_{\mathit{\text{relaxation}}}\left(t\right)=\left(E_1+E_2{e}^{\frac{-t}{{\tau }_R}}\right){\epsilon }_0$$(7) , the constitutive equation of the Maxwell model is needed, and its derivation can be found in Part F of the supplementary material.

The relaxation modulus as a function of time in the stress relaxation process can be obtained by dividing Eq. (7)(7) $${\sigma }_{\mathit{\text{relaxation}}}\left(t\right)=\left(E_1+E_2{e}^{\frac{-t}{{\tau }_R}}\right){\epsilon }_0$$(7) by ${\epsilon }_0:$ (8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) $E_{\mathit{\text{relaxation}}}\left(t\right)$ is a strictly decreasing function of $t$ at the interval $0\le t<\infty ,$ and please see Part H of the supplementary material for the proof.

From Eq. (8)(8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) , it can be observed that the relaxation modulus is $E_1+E_2$ at $t=0$ (here, $t=0$ is the time point at the beginning of the stress relaxation process), that is, $E_{\mathit{\text{relaxation}}}\left(0\right)=E_1+E_2,$ which is the maximal relaxation modulus in the stress relaxation process (please see Part H of the supplementary material for the proof). It can also be observed from Eq. (8)(8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) that the relaxation modulus is $E_1$ when $t\to \mathrm{\infty },$ that is, $\underset{t\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{relaxation}}}\left(t\right)=E_1,$ which is the minimal relaxation modulus in the stress relaxation process (please see Part H of the supplementary material for the proof). Hence, in the entire stress relaxation process from $t=0$ to $t\to \mathrm{\infty },$ the relaxation modulus strictly decreases over time by a total of $E_2$ (from $E_1+E_2$ at $t=0,$ to $E_1$ when $t\to \mathrm{\infty }$), and this time-dependent change must be due to the fluid part and is not relevant to the elastic solid part of a viscoelastic solid, since the fluid can flow and change with time while the elastic solid cannot, if the bulk deformation of the material is constant.

Once the fluid part gradually ceases to flow, the property of the fluid part gradually fades out, and the relaxation modulus approaches a constant $E_1$ and becomes time-independent when $t\to \mathrm{\infty }.$ Hence, the total amount of time-dependent decrease of the relaxation modulus $E_2$ in the entire stress relaxation process from $t=0$ to $t\to \mathrm{\infty }$ due to the fading of the property of the fluid part, must be the intrinsic modulus of elasticity of the fluid part of a viscoelastic solid. On the other hand, the property of the elastic solid part is time-independent and remains constant in the entire stress relaxation process. Once the fluid part ceases to flow and the intrinsic modulus of elasticity of the fluid part (that is, $E_2$) fades out when $t\to \mathrm{\infty },$ only the elastic solid part remains. Hence, the constant relaxation modulus when $t\to \mathrm{\infty },$ that is, $E_1,$ must be the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid. Thus far, the physical meanings of $E_1$ and $E_2$ of SLSM_Maxwell have been demonstrated based on the perspectives of the stress relaxation process.

Next, the demonstration based on the perspective of the creep process will be described below. Please note that, even though creep and stress relaxation look different, they are actually similar and interrelated, and can be regarded as the two sides of the same coin. Hence, the logic of the demonstration based on the creep process is similar to that based on the stress relaxation process described above, and therefore will be just briefly described below.

Using SLSM_Maxwell, the strain-time relationship of a viscoelastic solid in the creep process (that is, the solution of SLSM_Maxwell for describing the creep process) can be described by the equation: (9) $${\epsilon }_{\mathit{\text{creep}}}\left(t\right)=\frac{{\sigma }_0}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(9) where ${\epsilon }_{\mathit{\text{creep}}}\left(t\right)$ is the strain in the creep process as a function of time $t,$ and ${\sigma }_0$ is the constant stress in the creep process. Please see Part G of the supplementary material for the derivation of Eq. (9)(9) $${\epsilon }_{\mathit{\text{creep}}}\left(t\right)=\frac{{\sigma }_0}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(9) .

The creep compliance as a function of time in the creep process can be obtained by dividing Eq. (9)(9) $${\epsilon }_{\mathit{\text{creep}}}\left(t\right)=\frac{{\sigma }_0}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(9) by ${\sigma }_0:$ (10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) Using the relationship between creep compliance and relaxation modulus (that is, $E_{\mathit{\text{relaxation}}}\left(s\right)J_{\mathit{\text{creep}}}\left(s\right)=1∕s^2,$ where $s$ is the Laplace variable [132]), the creep compliance described by Eq. (10)(10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) can be converted to the description of the associated relaxation modulus in the creep process, which is equivalent to the relaxation modulus in the stress relaxation process described by Eq. (8)(8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) . Please see Part G of the supplementary material for the details.

From Eq. (10)(10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) , it can be observed that the creep compliance is $1∕\left(E_1+E_2\right)$ at $t=0$ (here, $t=0$ is the time point at the beginning of the creep process), that is, $J_{\mathit{\text{creep}}}\left(0\right)=1∕\left(E_1+E_2\right).$ Hence, $E_{\mathit{\text{relaxation}}}\left(0\right)=1∕J_{\mathit{\text{creep}}}\left(0\right)=E_1+E_2,$ which is the maximal relaxation modulus during the process. It can also be observed from Eq. (10)(10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) that the creep compliance is $1∕E_1$ when $t\to \mathrm{\infty },$ that is, $\underset{t\to \mathrm{\infty }}{\mathrm{\text{lim}}}{J}_{\mathit{\text{creep}}}\left(t\right)=1∕E_1.$ Hence, $\underset{t\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{relaxation}}}\left(t\right)=\underset{t\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[1∕J_{\mathit{\text{creep}}}\left(t\right)\right]=E_1,$ which is the minimal relaxation modulus during the process. In addition, it has been known that the relaxation modulus must be a strictly decreasing function of $t$ at the interval $0\le t<\infty .$ Hence, the logic of the rest demonstration is completely the same as that based on the perspective of the stress relaxation process described above.

Please see Part I of the supplementary material for the proof that $J_{\mathit{\text{creep}}}\left(t\right)$ is a strictly increasing function of $t$ at the interval $0\le t<\infty$ while $J_{\mathit{\text{creep}}}\left(0\right)=1∕\left(E_1+E_2\right)$ and $\underset{t\to \mathrm{\infty }}{\mathrm{\text{lim}}}{J}_{\mathit{\text{creep}}}\left(t\right)=1∕E_1$ are the minimal and maximal creep compliances in the creep process respectively.

Some simulation results for explaining the theoretical analysis proposed above are shown below. The simulation and data analysis are performed using MATLAB. The setting of the number and viscoelastic properties of materials in this simulation is the same as that of the simulation in the last subsection, as shown in Table 2. For each material, the relaxation modulus in the stress relaxation process is simulated using Eq. (8)(8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) while the associated relaxation modulus in the creep process is simulated using Eq. (10)(10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) , and their values at the beginning (at $t=0$) and at the end (when $t\to \mathrm{\infty }$) are recorded. It can be observed that the simulation results shown in Table 2 and Figure 5 can explain the theoretical analysis proposed above, that:

Figure 5. Simulation results associated with Table 2. The first column shows the relaxation modulus in the stress relaxation process simulated using Eq. (8)(8) $$E_{\mathit{\text{relaxation}}}\left(t\right)=\frac{{\sigma }_{\mathit{\text{relaxation}}}\left(t\right)}{{\epsilon }_0}=E_1+E_2{e}^{\frac{-t}{{\tau }_R}}$$(8) . The second column shows the creep compliance and associated relaxation modulus in the creep process simulated using Eq. (10)(10) $$J_{\mathit{\text{creep}}}\left(t\right)=\frac{{\epsilon }_{\mathit{\text{creep}}}\left(t\right)}{{\sigma }_0}=\frac{1}{E_1}\left(1-\frac{E_2}{E_1+E_2}{e}^{\frac{-t}{{\tau }_C}}\right)$$(10) . For each material, it can be observed that the relaxation modulus in the stress relaxation process in the first column is equivalent to the associated relaxation modulus in the creep process in the second column. (a) Results of materials with $E_1=5\mathrm{\text{kPa}}$ and $E_2=20\mathrm{\text{kPa}}.$ (b) Results of materials with $E_1=10\mathrm{\text{kPa}}$ and $E_2=40\mathrm{\text{kPa}}.$ (c) Results of materials with $E_1=30\mathrm{\text{kPa}}$ and $E_2=120\mathrm{\text{kPa}}.$

Table 2. Simulation results regarding the relaxation modulus in the stress relaxation process and the associated relaxation modulus in the creep process (those two relaxation moduli are actually equivalent) when $t=0$ and $t\to \infty ,$ for nine materials with different viscoelastic properties.

Material number	Setting of the viscoelastic properties			Relaxation modulus (Pa)
				Time point
	$E_1$ (kPa)	$E_2$ (kPa)	$\eta$ (kPa $·$ s)	$t=0$	$t\to \infty$
1	5	20	10	25,000	5000
2	5	20	40	25,000	5000
3	5	20	100	25,000	5000
4	10	40	20	50,000	10,000
5	10	40	80	50,000	10,000
6	10	40	200	50,000	10,000
7	30	120	60	150,000	30,000
8	30	120	240	150,000	30,000
9	30	120	600	150,000	30,000

1. The maximal relaxation modulus in the stress relaxation process is $E_1+E_2$ occurred at $t=0.$

2. The minimal relaxation modulus in the stress relaxation process is $E_1$ occurred when $t\to \infty .$

In summary, based on the perspectives of the stress relaxation and creep processes, the same conclusion can be reached regarding the physical meanings of $E_1$ and $E_2$ of SLSM_Maxwell: $E_1$ is the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid, and $E_2$ is the intrinsic modulus of elasticity of the fluid part of a viscoelastic solid.

3. Relationship and difference between the Maxwell and Kelvin forms of the standard linear solid model

In this section, the relationship and difference between SLSM_Maxwell and SLSM_Kelvin, and the physical meaning of each parameter of SLSM_Kelvin based on the perspective of the loading process will be demonstrated. Please note that, the logic of the demonstration of the physical meanings of the parameters for SLSM_Kelvin based on the perspectives of the stress relaxation and creep processes is similar to that for SLSM_Maxwell, therefore only the perspective of the loading process will be used for the demonstration here.

The constitutive equation of SLSM_Kelvin is: (11) $$\sigma +{\tau }_R^{\prime }\dot{\sigma }=E^{\prime }\left(\epsilon +{\tau }_C^{\prime }\dot{\epsilon }\right)$$(11) where ${\tau }_R^{\prime }={\eta }^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right)$ and ${\tau }_C^{\prime }={\eta }^{\prime }/E_2^{\prime }$ are the relaxation and creep time constants described by SLSM_Kelvin, respectively. $E^{\prime }$ is a constant equal to $E_1^{\prime }{E}_2^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right),$ while $E_1^{\prime },$ $E_2^{\prime }$ and ${\eta }^{\prime }$ are the parameters of SLSM_Kelvin relevant to the viscoelastic properties. Please see Part J of the supplementary material for the derivation of Eq. (11)(11) $$\sigma +{\tau }_R^{\prime }\dot{\sigma }=E^{\prime }\left(\epsilon +{\tau }_C^{\prime }\dot{\epsilon }\right)$$(11) .

By comparing Eqs. (1)(1) $$\sigma +{\tau }_R\dot{\sigma }=E_1\left(\epsilon +{\tau }_C\dot{\epsilon }\right)$$(1) and (11)(11) $$\sigma +{\tau }_R^{\prime }\dot{\sigma }=E^{\prime }\left(\epsilon +{\tau }_C^{\prime }\dot{\epsilon }\right)$$(11) , it can be observed that SLSM_Maxwell and SLSM_Kelvin have the same mathematical form of constitutive equation, therefore they have the same mathematical form of solution for describing every viscoelastic behavior (such as stress relaxation, creep, hysteresis, mechanical behavior in the loading process, and so on). It means that, although SLSM_Maxwell and SLSM_Kelvin are two different models, they are completely equivalent in terms of their mathematical properties as well as abilities for analysis and simulation (if ignoring the difference in the physical meanings of their parameters). In other words, they can produce completely consistent analysis and simulation results.

However, since SLSM_Maxwell and SLSM_Kelvin are two different models, $E_1$ of SLSM_Maxwell and $E_1^{\prime }$ of SLSM_Kelvin are two different parameters with different physical meanings, and therefore those two similarly-looking parameters are not equivalent. The same arguments can be made to $E_2$ and $E_2^{\prime },$ as well as to $\eta$ and ${\eta }^{\prime }.$ In other words, if SLSM_Maxwell and SLSM_Kelvin are simultaneously used to analyze the same set of viscoelastic data, the analysis results probably show that $E_1\ne E_1^{\prime },$ $E_2\ne E_2^{\prime }$ and $\eta \ne {\eta }^{\prime }.$ However, those two sets of analysis results, $\left(E_1,E_2,\eta \right)$ and $\left(E_1^{\prime },E_2^{\prime },{\eta }^{\prime }\right)$ obtained from SLSM_Maxwell and SLSM_Kelvin respectively, represent the same set of viscoelastic properties, since they are obtained from the same set of viscoelastic data determined by a specific set of viscoelastic properties. For example, if SLSM_Maxwell is used to analyze a set of viscoelastic data and the analysis result of $\left(E_1,E_2,\eta \right)=\left(5000,10,000,50,000\right)$ is obtained, it can be predicted that the corresponding $\left(E_1^{\prime },E_2^{\prime },{\eta }^{\prime }\right)$ must be $\left(15,000,7500,112,500\right)$ if SLSM_Kelvin is used to analyze the same set of viscoelastic data, based on the relationships between $\left(E_1,E_2,\eta \right)$ and $\left(E_1^{\prime },E_2^{\prime },{\eta }^{\prime }\right)$ shown below.

Let’s proceed to demonstrate the physical meaning of each parameter of SLSM_Kelvin. Using SLSM_Kelvin, the stress-strain relationship of a viscoelastic solid in the loading process (that is, the solution of SLSM_Kelvin for describing the loading process) can be described by the following equation, which can be derived from Eq. (11)(11) $$\sigma +{\tau }_R^{\prime }\dot{\sigma }=E^{\prime }\left(\epsilon +{\tau }_C^{\prime }\dot{\epsilon }\right)$$(11) : (12) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E^{\prime }\epsilon +E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-\epsilon }{r·{\tau }_R^{\prime }}}\right)$$(12) The derivation of Eq. (12)(12) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E^{\prime }\epsilon +E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-\epsilon }{r·{\tau }_R^{\prime }}}\right)$$(12) is almost identical to that of Eq. (2)(2) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E_1\epsilon +E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-\epsilon }{{r·\tau }_R}}\right)$$(2) , which can be found in Part D of the supplementary material.

The measured modulus of elasticity of a viscoelastic solid obtained through the loading process described by SLSM_Kelvin is: (13) $$E_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)-{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }_{t=0}\right)}{{\epsilon }^{\prime }-{\epsilon }_{t=0}}=\frac{{\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)}{{\epsilon }^{\prime }}=E^{\prime }+\frac{E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right)}{{\epsilon }^{\prime }}$$(13) where ${\sigma }_{\mathit{\text{loading}}}\left({\epsilon }^{\prime }\right)=E^{\prime }{\epsilon }^{\prime }+E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right),$ described using Eq. (12)(12) $${\sigma }_{\mathit{\text{loading}}}\left(\epsilon \right)=E^{\prime }\epsilon +E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-\epsilon }{r·{\tau }_R^{\prime }}}\right)$$(12) .

The measured modulus of elasticity when $r\to 0$ described by SLSM_Kelvin is: (14) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E^{\prime }+\frac{E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right)}{{\epsilon }^{\prime }}\right]=E^{\prime }=\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(14) The derivation of Eq. (14)(14) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E^{\prime }+\frac{E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right)}{{\epsilon }^{\prime }}\right]=E^{\prime }=\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(14) is almost identical to that of Eq. (4)(4) $${\underset{r\to 0}{\mathrm{\text{lim}}} E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to 0}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1$$(4) , which can be found in Part E of the supplementary material. Hence, the intrinsic modulus of elasticity of the elastic solid part of a viscoelastic solid described by SLSM_Kelvin is $E_1^{\prime }{E}_2^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right),$ which is the minimal measured modulus of elasticity that can be measured for a viscoelastic solid through the loading process occurred when $r\to 0.$

The measured modulus of elasticity when $r\to \mathrm{\infty }$ described by SLSM_Kelvin is: (15) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E^{\prime }+\frac{E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right)}{{\epsilon }^{\prime }}\right]=E_1^{\prime }$$(15) The derivation of Eq. (15)(15) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E^{\prime }+\frac{E^{\prime }r\left({\tau }_C^{\prime }-{\tau }_R^{\prime }\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{r·{\tau }_R^{\prime }}}\right)}{{\epsilon }^{\prime }}\right]=E_1^{\prime }$$(15) is almost identical to that of Eq. (5)(5) $$\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}{E}_{\mathit{\text{measured}}}\left({\epsilon }^{\prime }\right)=\underset{r\to \mathrm{\infty }}{\mathrm{\text{lim}}}\left[E_1+\frac{E_{1}r\left({\tau }_C-{\tau }_R\right)\left({1-e}^{\frac{-{\epsilon }^{\prime }}{{r·\tau }_R}}\right)}{{\epsilon }^{\prime }}\right]=E_1+E_2$$(5) , which can be found in Part E of the supplementary material. Hence, $E_1^{\prime }$ is the maximal measured modulus of elasticity that can be measured for a viscoelastic solid through the loading process occurred when $r\to \mathrm{\infty },$ described by SLSM_Kelvin.

The analysis in Subsection 2.2 has demonstrated the method for testing the intrinsic moduli of elasticity of the elastic solid part and fluid part of a viscoelastic solid through the loading process, that is, the intrinsic modulus of elasticity of the elastic solid part is equal to the measured modulus of elasticity tested when $r\to 0,$ while the intrinsic modulus of elasticity of the fluid part is equal to the difference between the measured moduli of elasticity tested when $r\to \mathrm{\infty }$ and $r\to 0.$ Hence, it can be deduced that the intrinsic modulus of elasticity of the fluid part of a viscoelastic solid described by SLSM_Kelvin is: (16) $$E_{\mathit{\text{maximal}}}-E_{\mathit{\text{minimal}}}=E_1^{\prime }-\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}=\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}$$(16) where $E_{\mathit{\text{maximal}}}$ and $E_{\mathit{\text{minimal}}}$ denote the maximal and minimal moduli of elasticity that can be measured for a viscoelastic solid through the loading process, respectively.

The demonstration above shows that, if SLSM_Kelvin is used to describe the viscoelastic properties of a viscoelastic solid, the intrinsic modulus of elasticity of the elastic solid part is $E_1^{\prime }{E}_2^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right),$ and the intrinsic modulus of elasticity of the fluid part is $E_1^{\prime 2}/\left(E_1^{\prime }+E_2^{\prime }\right).$ The physical meaning of $E_1^{\prime }$ of SLSM_Kelvin is the maximal measured modulus of elasticity that can be measured for a viscoelastic solid through the loading process.

Let’s do the following calculation to demonstrate the physical meaning of $E_2^{\prime }$ of SLSM_Kelvin: (17) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=\frac{\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}}{\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}}·E_1^{\prime }=E_2^{\prime }$$(17) where $E_{\mathit{\text{solid}}}$ and $E_{\mathit{\text{fluid}}}$ denote the intrinsic moduli of elasticity of the elastic solid part and fluid part of a viscoelastic solid, respectively. Equation (17)(17) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=\frac{\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}}{\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}}·E_1^{\prime }=E_2^{\prime }$$(17) suggests that the physical meaning of $E_2^{\prime }$ of SLSM_Kelvin is: $\left(E_{\mathit{\text{solid}}}∕E_{\mathit{\text{fluid}}}\right)·E_{\mathit{\text{maximal}}}.$

Next, the relationships between the parameters of SLSM_Maxwell and SLSM_Kelvin will be demonstrated. Even though SLSM_Maxwell and SLSM_Kelvin are two different models and their parameters are not equivalent, there exists relationships between their parameters.

It is important to know that no matter which model is used to analyze a set of viscoelastic data, the intrinsic viscoelastic properties of the analyzed material are unchanged. Hence, the following relationships hold: (18) $$E_{\mathit{\text{solid}}}=E_1=\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(18) (19) $$E_{\mathit{\text{fluid}}}=E_2=\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}$$(19) Equation (18)(18) $$E_{\mathit{\text{solid}}}=E_1=\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(18) suggests that $E_1$ is the intrinsic modulus of elasticity of the elastic solid part described by SLSM_Maxwell, and $E_1^{\prime }{E}_2^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right)$ is that described by SLSM_Kelvin. Equation (19)(19) $$E_{\mathit{\text{fluid}}}=E_2=\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}$$(19) suggests that $E_2$ is the intrinsic modulus of elasticity of the fluid part described by SLSM_Maxwell, and $E_1^{\prime 2}/\left(E_1^{\prime }+E_2^{\prime }\right)$ is that described by SLSM_Kelvin. Equations (18)(18) $$E_{\mathit{\text{solid}}}=E_1=\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(18) and (19)(19) $$E_{\mathit{\text{fluid}}}=E_2=\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}$$(19) describe the relationships between $E_1,$ $E_2,$ $E_1^{\prime }$ and $E_2^{\prime },$ expressing $E_1$ and $E_2$ as functions of $E_1^{\prime }$ and $E_2^{\prime }.$ Or, there is another point of view to describe the relationships between $E_1,$ $E_2,$ $E_1^{\prime }$ and $E_2^{\prime },$ by expressing $E_1^{\prime }$ and $E_2^{\prime }$ as functions of $E_1$ and $E_2:$ (20) $$E_{\mathit{\text{maximal}}}=E_1^{\prime }=E_1+E_2$$(20) (21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) Equation (20)(20) $$E_{\mathit{\text{maximal}}}=E_1^{\prime }=E_1+E_2$$(20) suggests that $E_1^{\prime }$ is the maximal measured modulus of elasticity described by SLSM_Kelvin, and $E_1+E_2$ is that described by SLSM_Maxwell. Equation (21)(21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) suggests that $E_2^{\prime }$ is $\left(E_{\mathit{\text{solid}}}/E_{\mathit{\text{fluid}}}\right)·E_{\mathit{\text{maximal}}}$ described by SLSM_Kelvin, and $\left(E_1/E_2\right)\left(E_1+E_2\right)$ is that described by SLSM_Maxwell.

The relationship between $\eta$ of SLSM_Maxwell and ${\eta }^{\prime }$ of SLSM_Kelvin can be described by the following equation: (22) $$\text{relaxation time constant}=\frac{\eta }{E_2}=\frac{{\eta }^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(22) where $\eta /E_2$ is the relaxation time constant described by SLSM_Maxwell, and ${\eta }^{\prime }/\left(E_1^{\prime }+E_2^{\prime }\right)$ is that described by SLSM_Kelvin. By substituting Eq. (19)(19) $$E_{\mathit{\text{fluid}}}=E_2=\frac{E_1^{\prime 2}}{E_1^{\prime }+E_2^{\prime }}$$(19) into Eq. (22)(22) $$\text{relaxation time constant}=\frac{\eta }{E_2}=\frac{{\eta }^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(22) , $\eta$ can be expressed as a function of $E_1^{\prime },$ $E_2^{\prime }$ and ${\eta }^{\prime }:$ (23) $$\eta ={\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$$(23) In addition, by substituting Eqs. (20)(20) $$E_{\mathit{\text{maximal}}}=E_1^{\prime }=E_1+E_2$$(20) and (21)(21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) into Eq. (22)(22) $$\text{relaxation time constant}=\frac{\eta }{E_2}=\frac{{\eta }^{\prime }}{E_1^{\prime }+E_2^{\prime }}$$(22) , ${\eta }^{\prime }$ can be expressed as a function of $E_1,$ $E_2$ and $\eta :$ (24) $${\eta }^{\prime }=\eta {\left(1+\frac{E_1}{E_2}\right)}^2$$(24)

Thus far, the relationships between the parameters of SLSM_Maxwell and SLSM_Kelvin have been established, as Eqs. (18)–(21)(21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) , (23)(23) $$\eta ={\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$$(23) , and (24)(24) $${\eta }^{\prime }=\eta {\left(1+\frac{E_1}{E_2}\right)}^2$$(24) .

It is important to note that Eq. (24)(24) $${\eta }^{\prime }=\eta {\left(1+\frac{E_1}{E_2}\right)}^2$$(24) suggests that the physical meaning of ${\eta }^{\prime }$ of SLSM_Kelvin is: ${{\eta }_{\mathit{\text{fluid}}}·\left(1+E_{\mathit{\text{solid}}}∕E_{\mathit{\text{fluid}}}\right)}^2,$ where ${\eta }_{\mathit{\text{fluid}}}$ denotes the viscosity of the fluid part of a viscoelastic solid. Hence, the physical meaning of ${\eta }^{\prime }$ of SLSM_Kelvin cannot be directly interpreted as the viscosity of the fluid part of a viscoelastic solid. In addition, it has been mentioned previously that the physical meaning of $\eta$ of SLSM_Maxwell is the viscosity of the fluid part of a viscoelastic solid. If this fact is set as the standard, Eq. (23)(23) $$\eta ={\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$$(23) suggests that the viscosity of the fluid part of a viscoelastic solid described by SLSM_Kelvin should be ${\eta }^{\prime }{\left[E_1^{\prime }∕\left(E_1^{\prime }+E_2^{\prime }\right)\right]}^2,$ not ${\eta }^{\prime }.$

Tables 3 and 4 summarize the physical meaning of each parameter of SLSM_Maxwell and SLSM_Kelvin, respectively. Tables 5 and 6 summarize the relationships between the physical properties of a viscoelastic solid and the parameters of SLSM_Maxwell and SLSM_Kelvin, respectively. It can be observed that, each parameter of SLSM_Maxwell has a significant and independent physical meaning relevant to the constitution of a viscoelastic solid. On the other hand, while the physical meaning of $E_1^{\prime }$ of SLSM_Kelvin is significant, it is not directly related to the constitution of a viscoelastic solid. It seems that the physical meanings of $E_2^{\prime }$ and ${\eta }^{\prime }$ of SLSM_Kelvin are not independent and straightforward.

Table 3. The physical meaning of each parameter of the Maxwell form of the standard linear solid model.

Parameter	Physical meaning
$E_1$	$E_{\mathit{\text{solid}}}$
$E_2$	$E_{\mathit{\text{fluid}}}$
$\eta$	${\eta }_{\mathit{\text{fluid}}}$

$E_{\mathit{\text{solid}}}$ and $E_{\mathit{\text{fluid}}}$ denote the intrinsic moduli of elasticity of the elastic solid part and fluid part of a viscoelastic solid, respectively. ${\eta }_{\mathit{\text{fluid}}}$ denotes the viscosity of the fluid part of a viscoelastic solid.

Table 4. The physical meaning of each parameter of the Kelvin form of the standard linear solid model.

Parameter	Physical meaning
$E_1^{\prime }$	$E_{\mathit{\text{maximal}}}$
$E_2^{\prime }$	$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}$
${\eta }^{\prime }$	${\eta }_{\mathit{\text{fluid}}}{\left(1+\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}\right)}^2$

$E_{\mathit{\text{solid}}}$ and $E_{\mathit{\text{fluid}}}$ denote the intrinsic moduli of elasticity of the elastic solid part and fluid part of a viscoelastic solid, respectively. $E_{\mathit{\text{maximal}}}$ denotes the maximal modulus of elasticity that can be measured for a viscoelastic solid through the loading process. ${\eta }_{\mathit{\text{fluid}}}$ denotes the viscosity of the fluid part of a viscoelastic solid.

Table 5. Relationships between the physical properties of a viscoelastic solid and the parameters of the Maxwell form of the standard linear solid model.

Physical property	Combination of parameters
Intrinsic modulus of elasticity of the elastic solid part	$E_1$
Intrinsic modulus of elasticity of the fluid part	$E_2$
Viscosity of the fluid part	$\eta$

Table 6. Relationships between the physical properties of a viscoelastic solid and the parameters of the Kelvin form of the standard linear solid model.

Physical property	Combination of parameters
Intrinsic modulus of elasticity of the elastic solid part	$\frac{E_1^{\prime }{E}_2^{\prime }}{E_1^{\prime }+E_2^{\prime }}$
Intrinsic modulus of elasticity of the fluid part	$\frac{{E_1^{\prime }}^2}{E_1^{\prime }+E_2^{\prime }}$
Viscosity of the fluid part	${\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$

The summary of this section will be provided in the first paragraph of the Discussion section below.

4. Discussion

The present paper demonstrates the physical meaning of each parameter of the Maxwell and Kelvin forms of the standard linear solid model, and demonstrates the relationship and difference between those two forms. Despite that SLSM_Maxwell and SLSM_Kelvin look similar, they are actually two different models and each parameter of them has its own specific physical meaning. It seems that the physical meanings of the parameters of SLSM_Maxwell are more significant. However, regardless of the physical significance of their parameters, SLSM_Maxwell and SLSM_Kelvin are completely equivalent in terms of their mathematical properties as well as abilities for analysis and simulation (since they have the same mathematical form of constitutive equation and the same mathematical form of solution for describing every viscoelastic behavior), and it means that they can produce completely consistent analysis and simulation results. Hence, no matter which model is chosen for analyzing and simulating viscoelastic behaviors, consistent results and accurate interpretations can always be obtained, as long as the following two points introduced in this paper are carefully followed. First, the physical meaning of each parameter of both models (as summarized in Tables 3 to 6) is accurately interpreted. Second, the relationships between the parameters of both models, that is, Eqs. (18)–(21)(21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) , (23)(23) $$\eta ={\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$$(23) , and (24)(24) $${\eta }^{\prime }=\eta {\left(1+\frac{E_1}{E_2}\right)}^2$$(24) , are accurately used for the transformations between those parameters.

In order to accurately evaluate the viscoelastic properties of materials as well as to accurately interpret the analysis results and optimize the application outcomes, it is important to clearly understand the physical meaning of each parameter of a viscoelastic model and to choose a model with parameters having significant physical meanings. By knowing the physical meaning of each parameter of a viscoelastic model, the viscoelastic meanings of the analysis results (that is, the numerical values of the parameters of the chosen viscoelastic model obtained from the analysis) as well as the state of the viscoelastic properties of the material being analyzed can be clearly understood. Having this valuable information, researchers can successfully carry out numerous applications and answer numerous research questions. For example, researchers can purposefully design and control the viscosity and intrinsic moduli of elasticity of the elastic solid part and fluid part of a material during designing and manufacturing processes by clearly understanding the states of those viscoelastic properties. In addition, researchers can purposefully tune the parameters that represent the viscoelastic properties during computational simulation, since researchers can clearly understand which parameter represents the viscosity while which parameters present the intrinsic moduli of elasticity of the elastic solid part and fluid part. If the parameters of a viscoelastic model have no significant physical meanings, the analysis results are just numerical values for analyzing the data while that model is just a purely mathematical model for data analysis without any physical meanings. Hence, in a study relevant to viscoelasticity, it is highly recommended to use a viscoelastic model with parameters having significant physical meanings.

In the viscoelastic models constructed using springs and dashpots, the generalized Maxwell and Kelvin models are considered to be the most general. However, the most major limitation of the generalized models is that they are not physically significant since their parameters do not have significant physical meanings. The other limitations of the generalized models, and some advantages of the standard linear solid model over the generalized models, have been discussed in the Introduction section. The present paper further demonstrates that the Maxwell form of the standard linear solid model is especially advantageous, because all of its parameters have significant physical meanings relevant to the constitution of a viscoelastic solid. However, the solution of the standard linear solid model for describing stress relaxation or creep is of the form of an exponential function plus a constant term, which is the characteristic of stress relaxation or creep of materials with simple compositions and structures tested using a tiny strain. Hence, only a limited number of stress relaxation and creep data can be successfully curvefitted by the standard linear solid model, since the stress relaxation and creep data of most materials should be mathematically described by two or more exponential functions plus a constant term. Fortunately, the standard linear solid model can be modified to be a general model that can account for the viscoelastic behaviors of most materials by replacing the exponential function in its solution form with a stretched-exponential function [133]. It has been reported that the stretched-exponential function is equivalent to a linear combination of exponential functions with arbitrary numbers [134–136], therefore the curvefitting ability of the modified standard linear solid model with stretched-exponential function is equivalent to that of the generalized models.

5. Conclusions

In conclusion, the present paper demonstrates the physical meaning of each parameter of the Maxwell and Kelvin forms of the standard linear solid model, and demonstrates the relationship and difference between those two forms. Despite that SLSM_Maxwell and SLSM_Kelvin look similar, they are actually two different models and each parameter of them has its own specific physical meaning. It seems that the physical meanings of the parameters of SLSM_Maxwell are more significant. However, regardless of the physical significance of their parameters, SLSM_Maxwell and SLSM_Kelvin are completely equivalent in terms of their mathematical properties as well as abilities for analysis and simulation, and it means that they can produce completely consistent analysis and simulation results. Hence, no matter which model is chosen for analyzing and simulating viscoelastic behaviors, consistent results and accurate interpretations can always be obtained, as long as the following two points introduced in this paper are carefully followed. First, the physical meaning of each parameter of both models (as summarized in Tables 3 to 6) is accurately interpreted. Second, the relationships between the parameters of both models, that is, Eqs. (18)–(21)(21) $$\frac{E_{\mathit{\text{solid}}}}{E_{\mathit{\text{fluid}}}}·E_{\mathit{\text{maximal}}}=E_2^{\prime }=\frac{E_1}{E_2}\left(E_1+E_2\right)$$(21) , (23)(23) $$\eta ={\eta }^{\prime }{\left(\frac{E_1^{\prime }}{E_1^{\prime }+E_2^{\prime }}\right)}^2$$(23) , and (24)(24) $${\eta }^{\prime }=\eta {\left(1+\frac{E_1}{E_2}\right)}^2$$(24) , are accurately used for the transformations between those parameters.

Disclosure statement

The author declares no conflict of interest relevant to this paper.

Additional information

Funding

The author acknowledges the financial support provided by the National Science and Technology Council for this paper (grant number: NSTC 111-2222-E-002-009-MY2).